Classification of mapping tori Assume $M$ is a topological space and $f\in \operatorname{Homeo}(M)$, then $f$ obviously plays a significant role in the torus bundle 
$$M_f = M\times I/\{(x,0)\sim (f(x),1)\mid x\in M\}.$$
Hence there should be some general result of the following type: $M_f$ and $M_g$ are bundle isomorphic (resp. diffeomorphic) if and only if "W", where W is a relation between $f$ and $g$.
Can someone help give W and explain? Thank you!
 A: Your $M_f$ is usually called the "mapping torus" of $f$, as Aaron points out.  It comes with a map to the circle $S^1 = I/(0 \sim 1)$ and this map is a fibre bundle with fibres isomorphic to $M$.  
First of all, the bundle isomorphism type of the bundle $M_f \to S^1$  only depends on the the isotopy class of $f$; i.e. if $f_0$ and $f_1$ are homotopic through homeomorphisms $f_t$ then $M_{f_0}$ and $M_{f_1}$ will be isomorphic bundles.  This reduces you to working with the mapping class group, which you probably already knew since I see the mapping-class-groups tag included.
Now, it's an easy exercise to check that $M_f$ and $M_g$ are isomorphic bundles if and only if $f$ and $g$ are conjugate in the mapping class group of $M$.
A: Mapping tori come equipped with a projection to the circle: $p\colon M_f\to S^1$. Bundle isomorphism is isomorphism of the total space that commutes with the projection. A weaker notion of isomorphism for mapping tori is an isomorphism that commutes with the projection only up to homotopy. Pseudoisotopy is the name for the relation on isomorphisms of $M$ that replaces isotopy in this coarser notion of equivalence of mapping tori. 
If $M$ is a simply connected manifold of high dimension, then pseudoisotopy implies isotopy and preserving the homotopy class of the map to the circle isn't a big deal, so isomorphism of mapping tori is pretty much isotopy of mapping classes. But if $M$ is not simply connected, there are two ways that mapping tori may be isomorphic without being bundle isomorphic: there may be more pseudoisotopies than isotopies and there may be isomorphisms that do not preserve the homotopy class of the map to the circle.
